openAI-math-dataset / lean /OAI /LinearAlgebra /MatrixMultiplication /Polynomial /ComplexPolynomialExecutionReindex.lean
| import OAI.LinearAlgebra.MatrixMultiplication.Polynomial.ComplexPolynomialKernelExecution | |
| namespace OAI | |
| /-! Polynomial tensor restrictions and exact coefficient extraction. -/ | |
| noncomputable section | |
| namespace MatrixMultiplication.Foundation.PolynomialKernelExecution.Execution | |
| open Tensor LocalMaps PolynomialLocalConstruction | |
| variable {X Y Z Prefix PX PY PZ AX AY AZ Prefix' PX' PY' PZ' : Type*} | |
| variable [Fintype AX] [Fintype AY] [Fintype AZ] | |
| variable {support : X → Y → Z → Prop} | |
| variable (E : Execution X Y Z Prefix PX PY PZ AX AY AZ support) | |
| variable (ePrefix : Prefix' ≃ Prefix) (eX : PX' ≃ PX) (eY : PY' ≃ PY) (eZ : PZ' ≃ PZ) | |
| def reindexOutput : Execution X Y Z Prefix' PX' PY' PZ' AX AY AZ support where | |
| auxiliary := E.auxiliary | |
| rankBound := E.rankBound | |
| auxiliary_rank := E.auxiliary_rank | |
| order := E.order | |
| leftDegree := E.leftDegree | |
| middleDegree := E.middleDegree | |
| rightDegree := E.rightDegree | |
| leftMap := fun x output input => E.leftMap x (ePrefix output.1, eX output.2) input | |
| middleMap := fun y output input => E.middleMap y (ePrefix output.1, eY output.2) input | |
| rightMap := fun z output input => E.rightMap z (ePrefix output.1, eZ output.2) input | |
| left_degree := fun x output input => | |
| E.left_degree x (ePrefix output.1, eX output.2) input | |
| middle_degree := fun y output input => | |
| E.middle_degree y (ePrefix output.1, eY output.2) input | |
| right_degree := fun z output input => | |
| E.right_degree z (ePrefix output.1, eZ output.2) input | |
| value := fun x y z => E.value | |
| (x.1, (ePrefix x.2.1, eX x.2.2)) | |
| (y.1, (ePrefix y.2.1, eY y.2.2)) | |
| (z.1, (ePrefix z.2.1, eZ z.2.2)) | |
| vanishes := fun x y z hs j hj => E.vanishes | |
| (x.1, (ePrefix x.2.1, eX x.2.2)) | |
| (y.1, (ePrefix y.2.1, eY y.2.2)) | |
| (z.1, (ePrefix z.2.1, eZ z.2.2)) hs j hj | |
| leading := fun x y z hs => E.leading | |
| (x.1, (ePrefix x.2.1, eX x.2.2)) | |
| (y.1, (ePrefix y.2.1, eY y.2.2)) | |
| (z.1, (ePrefix z.2.1, eZ z.2.2)) hs | |
| synchronized := by | |
| intro x y z hs hn | |
| have h := E.synchronized | |
| (x.1, (ePrefix x.2.1, eX x.2.2)) | |
| (y.1, (ePrefix y.2.1, eY y.2.2)) | |
| (z.1, (ePrefix z.2.1, eZ z.2.2)) hs hn | |
| exact ⟨ePrefix.injective h.1, ePrefix.injective h.2⟩ | |
| @[simp] theorem reindexOutput_auxiliary : | |
| (E.reindexOutput ePrefix eX eY eZ).auxiliary = E.auxiliary := rfl | |
| @[simp] theorem reindexOutput_rankBound : | |
| (E.reindexOutput ePrefix eX eY eZ).rankBound = E.rankBound := rfl | |
| @[simp] theorem reindexOutput_order : | |
| (E.reindexOutput ePrefix eX eY eZ).order = E.order := rfl | |
| @[simp] theorem reindexOutput_leftDegree : | |
| (E.reindexOutput ePrefix eX eY eZ).leftDegree = E.leftDegree := rfl | |
| @[simp] theorem reindexOutput_middleDegree : | |
| (E.reindexOutput ePrefix eX eY eZ).middleDegree = E.middleDegree := rfl | |
| @[simp] theorem reindexOutput_rightDegree : | |
| (E.reindexOutput ePrefix eX eY eZ).rightDegree = E.rightDegree := rfl | |
| @[simp] theorem reindexOutput_leftMap (x : X) (output : Prefix' × PX') (input : AX) : | |
| (E.reindexOutput ePrefix eX eY eZ).leftMap x output input = | |
| E.leftMap x (ePrefix output.1, eX output.2) input := rfl | |
| @[simp] theorem reindexOutput_middleMap (y : Y) (output : Prefix' × PY') (input : AY) : | |
| (E.reindexOutput ePrefix eX eY eZ).middleMap y output input = | |
| E.middleMap y (ePrefix output.1, eY output.2) input := rfl | |
| @[simp] theorem reindexOutput_rightMap (z : Z) (output : Prefix' × PZ') (input : AZ) : | |
| (E.reindexOutput ePrefix eX eY eZ).rightMap z output input = | |
| E.rightMap z (ePrefix output.1, eZ output.2) input := rfl | |
| @[simp] theorem reindexOutput_value | |
| (x : X × (Prefix' × PX')) (y : Y × (Prefix' × PY')) (z : Z × (Prefix' × PZ')) : | |
| (E.reindexOutput ePrefix eX eY eZ).value x y z = E.value | |
| (x.1, (ePrefix x.2.1, eX x.2.2)) | |
| (y.1, (ePrefix y.2.1, eY y.2.2)) | |
| (z.1, (ePrefix z.2.1, eZ z.2.2)) := rfl | |
| @[simp] theorem reindexOutput_kernel | |
| (x : X × (Prefix' × PX')) (y : Y × (Prefix' × PY')) (z : Z × (Prefix' × PZ')) : | |
| kernel (E.reindexOutput ePrefix eX eY eZ).auxiliary | |
| (E.reindexOutput ePrefix eX eY eZ).leftMap | |
| (E.reindexOutput ePrefix eX eY eZ).middleMap | |
| (E.reindexOutput ePrefix eX eY eZ).rightMap x y z = | |
| kernel E.auxiliary E.leftMap E.middleMap E.rightMap | |
| (x.1, (ePrefix x.2.1, eX x.2.2)) | |
| (y.1, (ePrefix y.2.1, eY y.2.2)) | |
| (z.1, (ePrefix z.2.1, eZ z.2.2)) := rfl | |
| theorem reindexOutput_fiberTransform | |
| (P : Tensor (Polynomial ℂ) (X × AX) (Y × AY) (Z × AZ)) | |
| (x : X × (Prefix' × PX')) (y : Y × (Prefix' × PY')) (z : Z × (Prefix' × PZ')) : | |
| fiberTransform (E.reindexOutput ePrefix eX eY eZ).leftMap | |
| (E.reindexOutput ePrefix eX eY eZ).middleMap | |
| (E.reindexOutput ePrefix eX eY eZ).rightMap P x y z = | |
| fiberTransform E.leftMap E.middleMap E.rightMap P | |
| (x.1, (ePrefix x.2.1, eX x.2.2)) | |
| (y.1, (ePrefix y.2.1, eY y.2.2)) | |
| (z.1, (ePrefix z.2.1, eZ z.2.2)) := rfl | |
| theorem reindexOutput_value_preimage | |
| (x : X × (Prefix × PX)) (y : Y × (Prefix × PY)) (z : Z × (Prefix × PZ)) : | |
| (E.reindexOutput ePrefix eX eY eZ).value | |
| (x.1, (ePrefix.symm x.2.1, eX.symm x.2.2)) | |
| (y.1, (ePrefix.symm y.2.1, eY.symm y.2.2)) | |
| (z.1, (ePrefix.symm z.2.1, eZ.symm z.2.2)) = E.value x y z := by | |
| simp only [reindexOutput_value, Equiv.apply_symm_apply] | |
| theorem reindexOutput_kernel_preimage | |
| (x : X × (Prefix × PX)) (y : Y × (Prefix × PY)) (z : Z × (Prefix × PZ)) : | |
| kernel (E.reindexOutput ePrefix eX eY eZ).auxiliary | |
| (E.reindexOutput ePrefix eX eY eZ).leftMap | |
| (E.reindexOutput ePrefix eX eY eZ).middleMap | |
| (E.reindexOutput ePrefix eX eY eZ).rightMap | |
| (x.1, (ePrefix.symm x.2.1, eX.symm x.2.2)) | |
| (y.1, (ePrefix.symm y.2.1, eY.symm y.2.2)) | |
| (z.1, (ePrefix.symm z.2.1, eZ.symm z.2.2)) = | |
| kernel E.auxiliary E.leftMap E.middleMap E.rightMap x y z := by | |
| simp only [reindexOutput_kernel, Equiv.apply_symm_apply] | |
| end MatrixMultiplication.Foundation.PolynomialKernelExecution.Execution | |
| end | |
| end OAI | |